j. differential geometry 63 (2003) 63-95

HARD LEFSCHETZ THEOREM FOR VALUATIONS, COMPLEX INTEGRAL GEOMETRY, AND UNITARILY INVARIANT VALUATIONS

SEMYON ALESKER

Abstract

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.

0. Introduction

In this paper we obtain new results on the structure of the space of even translation invariant continuous valuations on convex sets. In particular we prove a version of hard Lefschetz theorem for them and introduce certain natural duality operator which establishes an isomorphism between the space of such valuations on a linear space V and on its dual V^{∗} (with an appropriate twisting). Then we obtain an explicit geometric classification of unitarily invariant translation invariant continuous valuations on a Hermitian space ℂ^{n}. This classification is used to deduce new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.

Let us describe the results in more details. First let us remind the definition of valuation. Let V be a finite dimensional real vector space. Let K(V) denote the class of all convex compact subsets of V.

Received 08/05/2002.

Definition.

a) A function ϕ : K(V) ⟶ ℂ is called a valuation if for any K_{1}, K_{2} ∈ K(V) such that their union is also convex one has

ϕ(K_{1} ∪ K_{2}) = ϕ(K_{1}) + ϕ(K_{2}) − ϕ(K_{1} ∩ K_{2}).

b) A valuation ϕ is called continuous if it is continuous with respect the Hausdorff metric on K(V).

Remind that the Hausdorff metric d_{H} on K(V) depends on the choice of a Euclidean metric on V and it is defined as follows: d_{H}(A, B) := inf{ε > 0|A ⊂ (B)_{ε} and B ⊂ (A)_{ε}}, where (U)_{ε} denotes the ε-neighbor-hood of a set U. Then K(V) becomes a locally compact space (by the Blaschke selection theorem).

In this paper we are interested only in translation invariant continuous valuations. The space of such valuations will be denoted by Val (V). The simplest examples of such valuations are a Lebesgue measure on V and the Euler characteristic χ (which is equal to 1 on each convex compact set). For the classical theory of valuations we refer to the surveys [39], [40]. For a brief overview of more recent results see [3] and [4].

Definition. A valuation ϕ is called homogeneous of degree k (or k-homogeneous) if for every convex compact set K and for every scalar λ > 0

ϕ(λK) = λ^{k}ϕ(K).

Let us denote by Val_{k}(V) the space of translation invariant continuous k-homogeneous valuations.

Theorem (McMullen [38]).

Val (V) = ⨁_{k=0}^{n} Val_{k}(V),

where n = dim V.

In particular note that the degree of homogeneity is an integer between 0 and n = dim V. It is known that Val_{0}(V) is one-dimensional and it is spanned by the Euler characteristic χ, and Val_{n}(V) is also one-dimensional and is spanned by a Lebesgue measure [24]. The space Val_{n}(V) is also denoted by | ∧ V^{∗}| (or by Dens (V), the space of densities on V). Let us denote by Val^{ev}(V) the subspace of Val (V) of even valuations (a valuation ϕ is called even if ϕ(−K) = ϕ(K) for every K ∈ K(V)). Similarly one defines the subspace Val^{odd}(V) of odd valuations. One has further decomposition with respect to parity:

Val_{k}(V) = Val _{k}^{ev}(V) ⊕ Val _{k}^{odd}(V),

where Val _{k}^{ev}(V) is the subspace of even k-homogeneous valuations, and Val _{k}^{odd}(V) is the subspace of odd k-homogeneous valuations.

Let us fix on V a Euclidean metric, and let D denote the unit Euclidean ball with respect to this metric. Let us define on the space of translation invariant continuous valuations an operation Λ of mixing with the Euclidean ball D, namely

(Λϕ)(K) := \frac{d}{dε}|_{ε=0}ϕ(K + εD)

for any convex compact set K. Note that ϕ(K + εD) is a polynomial in ε ≥ 0 by McMullen’s theorem [38]. It is easy to see that the operator Λ preserves parity and decreases the degree of homogeneity by one. In particular we have

Λ : Val _{k}^{ev}(V) ⟶ Val _{k−1}^{ev}(V).

To formulate our first main result we will need one more definition from the representation theory. Let G be a Lie group. Let ρ be a continuous representation of G in a Fréchet space F. A vector v ∈ F is called G-smooth if the map G ⟶ F defined by g ⟼ g(v) is infinitely differentiable. It is well-known (and easy to prove) that smooth vectors form a linear G-invariant subspace which is dense in F. We will denote it by F^{sm}. It is well-known (see e.g., [49]) that F^{sm} has a natural structure of a Fréchet space, and the representation of G in F^{sm} is continuous with respect to this topology. In our situation the Fréchet space F = Val (V) with the topology of uniform convergence on compact subsets of K(V), and G = GL(V). The action of GL(V) on Val (V) is the natural one, namely for any g ∈ GL(V), ϕ ∈ Val (V) one has (g(ϕ))(K) = ϕ(g^{−1}K).

The following result is a version of the hard Lefschetz theorem.

Theorem 1.1.1. Let n/2 < k ≤ n. Then

Λ^{2k−n} : (Val _{k}^{ev}(V))^{sm} ⟶ (Val _{n−k}^{ev}(V))^{sm}

is an isomorphism. In particular for 1 ≤ i ≤ 2k − n the map

Λ^{i} : (Val _{k}^{ev}(V))^{sm} ⟶ (Val _{k−i}^{ev}(V))^{sm}

is injective.

Our terminology is motivated by the classical hard Lefschetz theorem (see e.g., [21]) about the cohomology of Kähler manifolds. To continue this analogy note that recently we have observed [5] the natural multiplicative structure on (Val (V))^{sm} (see also [4]). More precisely this space has natural structure of commutative associative graded algebra (where the grading is given by the degree of homogeneity). It satisfies a version on the Poincaré duality with respect to these multiplication and grading.

The operator Λ turns out to be closely related to so called cosine transform on real Grassmannians, and the proof of Theorem 1.1.1 is based on the solution of the cosine transform problem by J. Bernstein and the author [6] (some particular cases of this problem were solved previously by Matheron [37] and Goodey, Howard, and Reeder [19]).

Our next main result establishes connection between even translation invariant continuous valuations on V and on its dual space V^{∗}. In order to formulate it let us make an elementary remark from linear algebra. Let E ⊂ V be any k-dimensional subspace. One has the canonical isomorphism | ∧^{n} V | = | ∧^{k} E| ⊗ | ∧^{n−k} (V/E)|. Note also that V/E = (E^{⊥})^{∗}. Hence we get the canonical isomorphism

| ∧^{k} E^{∗}| = | ∧^{n−k} (E^{⊥})^{∗}| ⊗ | ∧^{n} V^{∗}|.

Then we have:

Theorem 1.2.1. For any k = 0, 1, …, n(= dim V) there exists a natural isomorphism

𝔻 : (Val _{k}^{ev}(V))^{sm} ˜⟶ (Val _{n−k}^{ev}(V^{∗}))^{sm} ⊗ | ∧^{n} V^{∗}|.

This isomorphism 𝔻 is uniquely characterized by the following property: let ϕ ∈ Val _{k}^{ev}(V) and let E ∈ Gr_{k}(V); then ϕ|_{E} = 𝔻(ϕ)|_{E^{⊥}} under the above identification | ∧^{k} E^{∗}| = | ∧^{n−k} (E^{⊥})^{∗}| ⊗ | ∧^{n} V^{∗}|.

The proof of this theorem uses the representation theoretical interpretation of the space Val^{ev}(V) given in [2], where this space was characterized as the unique irreducible submodule of some standard GL(V)-module with smallest Gelfand-Kirillov dimension (of the corresponding Harish-Chandra module).

Now let us discuss the translation invariant continuous valuations invariant under some group G of linear transformations of V. This space will be denoted by Val^{G}(V). If G is the group of orthogonal transformations O(n) or special orthogonal transformations SO(n) the corresponding space of valuations is described explicitly by the following famous result of H. Hadwiger.

Theorem (Hadwiger, [24]). Let V be n-dimensional Euclidean space. The intrinsic volumes V_{0}, V_{1}, …, V_{n} form a basis of Val^{SO(n)}(V) (= Val^{O(n)}(V)).

Let us remind the definition of the intrinsic volumes V_{i}. Let Ω be a compact (not necessarily convex) domain in a Euclidean space V with smooth boundary ∂Ω. Let n = dim V. For any point s ∈ ∂Ω let k_{1}(s), …, k_{n−1}(s) denote the principal curvatures at s. For 0 ≤ i ≤ n−1 define

V_{i}(Ω) := \frac{1}{(n − i)vol_{n−i}(D_{n−i})}\binom{n − 1}{n − 1 − i}^{−1} ∫_{∂Ω}\{k_{j_{1}}, …, k_{j_{n−1−i}}\}dσ,

where {k_{j_{1}}, …, k_{j_{n−1−i}}} denotes the (n−1−i)-th elementary symmetric polynomial in the principal curvatures, dσ is the measure induced on ∂Ω by the Euclidean metric, and D_{n−i} denotes the unit (n − i)-dimensional ball. It is well-known (see e.g., [44]) that V_{i} (uniquely) extends by continuity in the Hausdorff metric to K(V). Define also V_{n}(Ω) := vol (Ω). Note that V_{0} is proportional to the Euler characteristic χ. It is wellknown that V_{0}, V_{1}, …, V_{n} belong to Val^{O(n)}(V). It is easy to see that V_{k} is homogeneous of degree k.

Now let us describe unitarily invariant valuations on the Hermitian space ℂ^{n}. Let us denote by IU(n) the group of isometries of the Hermitian space ℂ^{n} preserving the complex structure (thus IU(n) = ℂ^{n}⋊U(n)). Let^{C}AGr_{j} denote the Grassmannian of affine complex subspaces of ℂ^{n} of complex dimension j. Clearly ^{C}AGr_{j} is a homogeneous space of IU(n) and it has a unique (up to a constant) IU(n)-invariant measure (called Haar measure). For every nonnegative integers p and k such that 2p ≤ k ≤ 2n let us introduce the following valuations:

U_{k,p}(K) = ∫_{E∈^{C}AGr_{n−p}} V_{k−2p}(K ∩ E) · dE.

Then U_{k,p} ∈ Val _{k}^{U(n)}(ℂ^{n}).

Theorem 2.1.1. The valuations U_{k,p} with 0 ≤ p ≤ \frac{min\{k,2n−k\}}{2} form a basis of the space Val _{k}^{U(n)}(ℂ^{n}).

This result is the Hermitian generalization of the (Euclidean) Hadwiger theorem. The proof of this theorem is highly indirect. It turns out to be necessary to study the GL_{2n}(ℝ)-module structure of the infinite dimensional space Val^{ev}(ℂ^{n}). The proof of Theorem 2.1.1 uses most of the facts known about even valuations including the solution of the McMullen conjecture [2], cosine transform [6], the hard Lefschetz theorem for valuations, and the results of Howe and Lee [26] on the K-type structure of certain GL-modules.

Note that there are some other natural examples of valuations from Val^{U(n)}(ℂ^{n}), for instance the averaged volume of projections of a convex set to all complex (or, say, Lagrangian) subspaces. Theorem 2.1.1 implies that all of them are linear combinations of U_{k,p} with the above range of indices k, p. We would also like to mention another interesting example of such valuation which comes from the complex analysis. It is so called Kazarnovskii’s pseudovolume. It was introduced and studied by B. Kazarnovskii [30], [31] in order to write down a formula for the number of zeros of a system of exponential sums in terms of their Newton polytopes. His results generalize in some sense the well-known results of D. Bernstein [7] and A. Kouchnirenko [35] on the number of zeros of a system of polynomial equations (see also [18]). We will recall the definition of Kazarnovskii’s pseudovolume in Subsection 3.3. As a corollary of Theorem 2.1.1 we present a new formula for Kazarnovskii’s pseudovolume in integral geometric terms (Theorem 3.3.2). It also seems that the valuation property of Kazarnovskii’s pseudovolume was not mentioned previously in the literature.

The classification of unitarily invariant valuations is used to obtain new integral geometric formulas in the Hermitian space ℂ^{n}. Let us state some of them. Let Ω_{1}, Ω_{2} be compact domains with smooth boundary in ℂ^{n} such that Ω_{1}∩U(Ω_{2}) has finitely many components for all U ∈ IU(n). The new result is:

Theorem 3.1.1. Let Ω_{1}, Ω_{2} be compact domains in ℂ^{n} with piecewise smooth boundaries such that for every U ∈ IU(n) the intersection Ω_{1} ∩ U(Ω_{2}) has finitely many components. Then

∫_{U∈IU(n)} χ(Ω_{1} ∩ U(Ω_{2}))dU

= ∑_{k_{1}+k_{2}=2n} ∑_{p_{1},p_{2}} κ(k_{1}, k_{2}, p_{1}, p_{2})U_{k_{1},p_{1}}(Ω_{1})U_{k_{2},p_{2}}(Ω_{2}),

where the inner sum runs over 0 ≤ p_{i} ≤ k_{i}/2, i = 1, 2, and κ(k_{1}, k_{2}, p_{1}, p_{2}) are certain uniquely defined constants depending on n, k_{1}, k_{2}, p_{1}, p_{2} only.

The study of the left-hand side in this formulas was started by J. Fu [16].

Theorem 3.1.2. Let Ω be a compact domain in ℂ^{n} with piecewise smooth boundary. Let 0 < q < n, 0 < 2p < k < 2q. Then

∫_{E∈^{C}AGr_{q}} U_{k,p}(Ω ∩ E) = ∑_{p=0}^{[k/2]+n−q} γ_{p} · U_{k+2(n−q),p}(Ω),

where the constants γ_{p} depend only on n, q, and p.

Let us denote by ALGr(ℂ^{n}) the (noncompact) Grassmannian of affine Lagrangian subspaces of ℂ^{n}. Clearly it is a homogeneous space of the group IU(n) and hence has a Haar measure.

Theorem 3.1.3. Let Ω be a compact domain in ℂ^{n} with piecewise smooth boundary. Then

∫_{ALGr(ℂ^{n})} χ(E ∩ Ω)dE = ∑_{p=0}^{[n/2]} β_{p} · U_{n,p}(Ω),

where β_{p} are certain uniquely defined constants depending on n and p only.

Theorems 3.1.1 and 3.1.2 are analogs of general kinematic formulas of Chern [11], [13] and Federer [15] (see also [43], especially Ch. 15, and [34]). Further generalizations in the Euclidean case were obtained by Cheeger, Müller, and Schrader [10] and J. Fu [16]. For more recent results in this direction and further references we refer to the recent survey by Hug and Schneider [27]. For classical results in Hermitian integral geometry we refer to [12], [20], [42]. In these papers the authors discuss the integral geometry of complex submanifolds. The integral geometry of Lagrangian submanifolds also was studied (see e.g., [36]). The integral geometry of real submanifolds in the complex projective space ℂP^{n} was studies by H. Tasaki [47], [48] and Kang and Tasaki [28], [29]. In these papers the authors obtain explicit Poincaré type formulas for real submanifolds of certain specific dimensions. Their results use in turn a general Poincaré type formula in Riemannian homogeneous spaces due to R. Howard [25].

It would be of interest to compute the constants κ(k_{1}, k_{2}, p_{1}, p_{2}), γ_{p}, and β_{p} in Theorems 3.1.1, 3.1.2, and 3.1.3 explicitly. We could not do it in general. But we were able to compute them only in the first nontrivial case n = 2. One of such computations is presented in Subsection 3.2. Much more complete treatment of the integral geometric formulas (including computation of all constants) in ℂ^{2}, ℂ^{3}, and also in the 2- and 3-dimensional complex projective and hyperbolic spaces was done recently by H. Park in his thesis [41].

The paper is organized as follows. In Section 1 we discuss the results about the structure of the space of even translation invariant continuous valuations. Namely in Subsection 1.1 we prove the hard Lefschetz theorem for valuations and deduce some corollaries from it. In Subsection 1.2 we discuss the duality on valuations, in particular we prove Theorem 1.2.1. In Section 2 we prove the classification of unitarily invariant translation invariant continuous valuations. In Section 3 we discuss the integral geometry in complex spaces. In Subsection 3.1 we obtain the integral geometric formulas in ℂ^{n}. In Subsection 3.2 we compute explicitly the constants in one of such formulas in ℂ^{2} (Theorem 3.2.4).

Acknowledgements. We express our gratitude to J. Bernstein for numerous useful discussions. We would also like to thank P. Biran, J. Fu, and L. Polterovich for useful conversations.

1. Hard Lefschetz theorem and duality for valuations

Let V be an n-dimensional real vector space. In Subsection 1.1 of this section we prove an analogue of the hard Lefschetz theorem for translation invariant even continuous valuations. In Subsection 1.2 we introduce the notion of a valuation dual to a given translation invariant even continuous valuation which satisfies some additional mild technical condition of GL(V)- smoothness (defined in the introduction). This construction uses the representation theoretical interpretation of the space of valuations given in [2]. The geometric examples will be given in Proposition 2.1.7 of Section 2.

1.1 An analogue of the hard Lefschetz theorem for valuations

The main result of this subsection is the following analogue of the hard Lefschetz theorem where the operator Λ was defined in the introduction.

Theorem 1.1.1. Let n ≥ k > n/2. Then Λ^{2k−n} : (Val _{k}^{ev}(V))^{sm} ⟶ (Val _{n−k}^{ev}(V))^{sm} is an isomorphism. In particular Λ^{i} : Val _{k}^{ev}(V) ⟶ Val _{k−i}^{ev}(V) is injective for 1 ≤ i ≤ 2n − k.

The proof of this theorem uses the cosine transform on real Grassmannians, thus we will remind first its definition and the relevant properties. We will denote by ^{R}Gr_{j}(V) the Grassmannian of real j-dimensi-onal linear subspaces in V. Assume that 1 ≤ i ≤ j ≤ n − 1. For two subspaces E ∈ ^{R}Gr_{i}(V), F ∈ ^{R}Gr_{j}(V) let us define the cosine of the angle between E and F:

| cos(E, F)| := \frac{vol_{i}(Pr_{F} (A))}{vol_{i}(A)},

where A is any subset of E of nonzero volume, Pr_{F} denotes the orthogonal projection onto F, and vol_{i} is the i-dimensional measure induced by the Euclidean metric. (Note that this definition does not depend on the choice of a subset A ⊂ E). In the case i ≥ j we define the cosine of the angle between E and F as cosine of the angle between their orthogonal complements:

| cos(E, F)| := | cos(E^{⊥}, F^{⊥})|.

(It is easy to see that for i = j both definitions coincide.)

For any 1 ≤ i, j ≤ n − 1 one defines the cosine transform

T_{j,i} : C(^{R}Gr_{i}(V)) ⟶ C(^{R}Gr_{j}(V))

as follows:

(T_{j,i}f)(F) := ∫_{^{R}Gr_{i}(V)} | cos(E, F)|f(E)dE,

where the integration is with respect to the Haar measure on the Grassmannian such that the total measure is equal to 1. Clearly the cosine transform commutes with the action of the orthogonal group O(n), and hence its image is an O(n)-invariant subspace of functions.

Now let us recall the imbedding Val _{k}^{ev}(V) ⟶ C(^{R}Gr_{k}(V)) which we will call the Klain imbedding. Let ϕ ∈ Val _{k}^{ev}(V). For every E ∈ RGr_{k}(V) let us consider the restriction of ϕ to all convex compact subsets of E. This is an even translation invariant valuation homogeneous of degree k. Hence, by a result due to Hadwiger [24], it is a density on E (i.e., a Lebesgue measure). Thus it is equal to f(E) · vol_{E}, where vol_{E} is the volume form on E defined by the metric on V, and f(E) is a constant depending on E. Thus ϕ ↦ f defines the map Val _{k}^{ev}(V) ⟶ C(^{R}Gr_{k}(V)) which turns out to be an imbedding by a result due to D. Klain ([32]; this result was stated in this form in [33] and in [1]). Let us denote this image by I_{k}. Moreover it was shown in [6] that the image of the Klain imbedding coincides with the image of the cosine transform T_{k,k} : C(^{R}Gr_{k}(V)) ⟶ C(^{R}Gr_{k}(V)) (at least on the level of GL(V)-smooth vectors).

Lemma 1.1.2. Let k ≥ n/2. The cosine transform

T_{n−k,k} : C(^{R}Gr_{k}(V)) ⟶ C(^{R}Gr_{n−k}(V))

maps I_{k} to I_{n−k} and induces isomorphism of O(n)-smooth vectors of these subspaces.

Proof. It is well-known that for admissible GL(V)-modules of finite length the subspaces of GL(V)-smooth and O(n)-smooth vectors coincide (more generally, GL(V) can be replaced by any real reductive group G, and O(n) can be replaced by a maximal compact subgroup of G). First let us prove that I_{k} and I_{n−k} have the same decomposition under the action of the orthogonal group O(n). Indeed the correspondence E ↦ E^{⊥} induces an isomorphism S : C(^{R}Gr_{k}(V)) ⟶ C(^{R}Gr_{n−k}(V)) commuting with the action of O(n). Moreover we have the following relation between the cosine transforms:

T_{n−k,n−k} = ST_{k,k}S^{−1}.

Hence it follows that S((I_{k})^{sm}) = (I_{n−k})^{sm} (it is immediate on the level of O(n)-finite vectors; to deduce it for O(n)-smooth vectors one should use the Casselman-Wallach theorem [9] as it is done in [6]).

Next it is well-known (see e.g., [6], Lemma 1.7) that the cosine transform T_{n−k,k} can be written (up to a nonzero normalizing constant which we ignore) as a composition T_{n−k,n−k} ∘ R_{n−k,k}, where R_{n−k,k} : C(^{R}Gr_{k}(V)) ⟶ C(^{R}Gr_{n−k}(V)) is the Radon transform. It was shown in [17] that

R_{n−k,k} : C^{∞}(^{R}Gr_{k}(V)) ⟶ C^{∞}(^{R}Gr_{n−k}(V))

is an isomorphism. We claim that R_{n−k,k}((I_{k})^{sm}) = (I_{n−k})^{sm}. To see this remind that the quasiregular representation of O(n) in the space of functions on the Grassmannians is multiplicity free (since the Grassmannians are symmetric spaces). Hence it follows that two O(n)-invariant closed subspaces of C^{∞}(^{R}Gr_{n−k}(V)) have the same O(n)-finite vectors if and only if these subspaces have the same decomposition under the action of O(n) (in the abstract sense). Hence R_{n−k,k}(I_{k}) and I_{n−k} have the same O(n)-finite vectors. The coincidence of O(n)-smooth vectors follows again from the Casselman-Wallach theorem [9] and the fact that the Radon transform can be rewritten as an intertwining operator of admissible GL(V)-modules of finite length (see [17]).

Since T_{n−k,n−k} is selfadjoint its restriction to I_{n−k} has trivial kernel and dense image. But the key observation of [6] was that T_{n−k,n−k} can be rewritten as an intertwining operator of certain GL(V)-modules. This and the Casselman-Wallach theorem [9] imply that

T_{n−k,n−k}((I_{n−k})^{sm}) = (I_{n−k})^{sm}.

Hence T_{n−k,k}((I_{k})^{sm}) = (I_{n−k})^{sm}. q.e.d.

Now let us prove Theorem 1.1.1.

Proof of Theorem 1.1.1. Since the image of GL(V)-smooth continuous k-homogeneous valuations in C(^{R}Gr_{k}(V)) coincides with the image of the cosine transform on GL(V)-smooth functions, then every GL(V)-smooth valuation ϕ ∈ Val _{k}^{ev}(V) can be represented in the form

ϕ(K) = ∫_{^{R}Gr_{k}(V)} f(E)vol_{k}(Pr_{E}(K))dE,

where f is a smooth function on ^{R}Gr_{k}(V), K is an arbitrary convex compact set, Pr_{E} denotes the orthogonal projection onto E, and the integration is with respect to the Haar measure on the Grassmannian. Moreover for every smooth function f, the expression defined by this formula is a valuation from (Val _{k}^{ev}(V))^{sm}. For a given valuation ϕ the function f is not defined uniquely. But we can choose f ∈ I_{k}, i.e., in the image of the cosine transform; then it will be defined uniquely. So we will assume that f ∈ I_{k}. Let us apply Λ^{2k−n} to it. Then it is easy to see that

(Λ^{2k−n}ϕ)(K) = c · ∫_{^{R}Gr_{k}(V)} f(E)V_{n−k}(Pr_{E}(K))dE,

where c is a nonzero normalizing constant, and V_{n−k}(Pr_{E}(K)) denotes the (n − k)-th intrinsic volume of Pr_{E}(K) inside E, i.e., it is the mixed volume of Pr_{E}(K) taken n − k times with the unit ball of E taken 2k − n times. The image g of Λ^{2k−n}ϕ in functions on the Grassmannian C(^{R}Gr_{n−k}(V)) can be described as follows. It is easy to see that for every subspace F ∈ ^{R}Gr_{n−k}(V)

g(F) = c′ · ∫_{^{R}Gr_{n−k}(V)} f(E)| cos(F, E)|dE,

where c′ is a nonzero normalizing constant. Namely g is equal (up to a normalization) to the cosine transform T_{n−k,k}(f) of f. By Lemma 1.1.2 T_{n−k,k} induces the isomorphism between GL(V)-smooth vectors of I_{k} and of I_{n−k}. This proves Theorem 1.1.1. q.e.d.

For a subgroup G ⊂ GL(V) let us denote by Val _{k}^{G}(V) the space of translation invariant G-invariant k-homogeneous continuous valuations. Let h_{k} := dim Val _{k}^{G}(V).

Corollary 1.1.3. Let G be a compact subgroup of the orthogonal group which acts transitively on the unit sphere and contains the operator −Id. Then Val _{k}^{G}(V) is a finite dimensional space, and for n/2 < k ≤ n

Λ^{2k−n} : Val _{k}^{G}(V) ⟶ Val _{n−k}^{G}(V)

is an isomorphism. Consequently the numbers h_{i} satisfy the Lefschetz inequalities:

h_{i} ≤ h_{i+1} for i < n/2, and h_{i} = h_{n−i} for i = 0, …, n.

Proof. The finite dimensionality of Val _{k}^{G}(V) was proved in [1]. Let us show that this implies that all vectors from Val _{k}^{G}(V) are O(n)-finite (in particular GL(V)-smooth). Indeed let Z be the minimal closed O(n)- invariant subspace of the space Val_{k}(V) containing Val _{k}^{G}(V). The space Z is decomposed under the action of O(n) into the direct sum of irreducible components, and each component enters with finite multiplicity (since the space of translation invariant continuous valuations of the given degree of homogeneity and parity can be realized as a subquotient of a representation of GL(V) induced from a character of a parabolic subgroup, see Section 2 in [2]). Thus let Z = ⊕_{i}ρ_{i} be this decomposition. We have a continuous projection π : Val_{k}(V) ⟶ Val _{k}^{G}(V) defined by π(ϕ) = ∫_{g∈G} g(ϕ)dg. Clearly Im(π) = Val _{k}^{G}(V) = (Z)^{G}. But (Z)^{G} = ⊕_{i}(ρ_{i})^{G}. Since Val _{k}^{G}(V) is finite dimensional, (ρ_{i})^{G} = 0 for all but finitely many i’s. In other words there is a finite set of indices A such that Val _{k}^{G}(V) ⊂ ⊕_{i∈A}ρ_{i}. Thus all elements of Val _{k}^{G}(V) are O(n)-finite.

Next obviously Λ(Val _{k}^{G}(V)) ⊂ Val _{k−1}^{G}(V). The rest follows from Theorem 1.1.1. q.e.d.

1.2 Duality on valuations

Let V be an n-dimensional real vector space. Let us denote by V^{∗} its dual space. Let us denote by | ∧^{n} V^{∗}| the (one-dimensional) space of complex-valued Lebesgue measures on V. Let us consider the space Val _{k}^{ev}(V^{∗}) ⊗ | ∧^{n} V^{∗}| of translation invariant even continuous k-homo-geneous valuations on V^{∗} with values in | ∧^{n} V^{∗}|. Note that on both spaces we have the natural (continuous) representation of the group GL(V).

Before we state the main result of this subsection let us make a remark. For any subspace E ∈ Gr_{k}(V) consider the short exact sequence 0 ⟶ E ⟶ V ⟶ V/E ⟶ 0. From this sequence one gets the canonical isomorphism | ∧^{n} V | = | ∧^{k} E| ⊗ | ∧^{n−k} (V/E)|. Note also that V/E = (E^{⊥})^{∗}. Hence we get the canonical isomorphism

| ∧^{k} E^{∗}| = | ∧^{n−k} (E^{⊥})^{∗}| ⊗ | ∧^{n} V^{∗}|.

The main result of this subsection is:

Theorem 1.2.1. For any k = 0, 1, …, n there exists a natural isomorphism

𝔻 : (Val _{k}^{ev}(V))^{sm} ˜⟶ (Val _{n−k}^{ev}(V^{∗}))^{sm} ⊗ | ∧^{n} V^{∗}|.

This isomorphism 𝔻 is defined uniquely by the following property: let ϕ ∈ (Val _{k}^{ev}(V))^{sm} and let E ∈ Gr_{k}(V); then ϕ|_{E} = 𝔻(ϕ)|_{E^{⊥}} under the above identification | ∧^{k} E^{∗}| = | ∧^{n−k} (E^{⊥})^{∗}| ⊗ | ∧^{n} V^{∗}|.

Proof. First let us rewrite the Klain imbedding we have discussed in Subsection 1.1 of the space of even valuations Val _{k}^{ev}(V) to functions on the Grassmannian in the notation which does not use any Euclidean structure. Instead of functions on the Grassmannian we have to consider sections of certain line bundle L_{k} over the Grassmannian^{R}Gr_{k}(V). The fiber of L_{k} over a subspace E ∈ ^{R}Gr_{k}(V) is the (one-dimensional) space |∧^{k} E^{∗}| of complex valued Lebesgue measures on E. Clearly L_{k} is naturally GL(V)-equivariant. Let us denote by C(^{R}Gr_{k}, L_{k}) the space of continuous sections of L_{k}. The map we have described in Subsection 1.1 can be rewritten as follows. Fix a valuation ϕ ∈ Val _{k}^{ev}(V). For any E ∈ ^{R}Gr_{k}(V) let us consider the restriction of ϕ to E. As previously, since this restriction ϕ|_{E} has maximal degree of homogeneity (equal to k) by Hadwiger’s theorem [24] ϕ|_{E} is a Lebesgue measure on E. Thus ϕ defines a continuous section of L_{k}. As we have mentioned, the constructed map is injective. One of the main results of [2] says that the image of Val _{k}^{ev}(V) in C(^{R}Gr_{k}(V), L_{k}) under this map is the unique “small” irreducible GL(V)-submodule (Theorem 1.3 combined with Theorem 3.1 in [2]) . Let us give some comments what does it mean “small”. First replace all GL(V)-modules by their Harish-Chandra modules which are purely algebraic objects. For each Harish-Chandra module one defines an associated variety (or Bernstein’s variety) which is an algebraic subvariety of the Lie algebra gl_{n}(ℂ), where n = dim(V). For details we refer to [8]. When we say that a given irreducible submodule A of a module B is “small” it means that the dimensions of the associated varieties of all other irreducible subquotients of B are strictly greater than that of A. Note also that the dimension of the associated variety of A is equal to the Gelfand-Kirillov dimension of the underlying Harish-Chandra module.

Now let us continue constructing the isomorphism 𝔻. Let us consider the line bundle M_{k} over ^{R}Gr_{n−k}(V^{∗}) the fiber of which over any F ∈ ^{R}Gr_{n−k}(V^{∗}) is equal to | ∧^{n−k} F^{∗}| ⊗ | ∧^{n} V^{∗}| (note that | ∧^{n−k} F^{∗}| is identified with the space of Lebesgue measures on F). As previously, Val _{n−k}^{ev}(V^{∗}) ⊗ | ∧^{n} V^{∗}| can be realized as the only “small” irreducible submodule of C(^{R}Gr_{n−k}(V^{∗}), M_{k}) (indeed these spaces differ from the previous two only by the twist by | ∧^{n} V^{∗}|). Hence it is sufficient to present the natural isomorphism between C^{∞}(^{R}Gr_{k}(V), L_{k}) and C^{∞}(^{R}Gr_{n−k}(V^{∗}), M_{k}) where C^{∞} denotes the space of C^{∞}-sections of the bundles. Let us do it. Let E ∈ ^{R}Gr_{k}(V). As previously, we have the canonical isomorphism

| ∧^{k} E^{∗}| = | ∧^{n−k} (E^{⊥})^{∗}| ⊗ | ∧^{n} V^{∗}|.

The correspondence E ⟼ E^{⊥} and the last identification give the desired isomorphism. q.e.d.

Now let us assume that V is a Euclidean space, i.e., on V we are given a positive definite quadratic form. This gives us the identification of V with its dual space V^{∗}, and the identification of the space | ∧^{n} V^{∗}| of Lebesgue measures on V with the complex line ℂ (such that 1 ∈ ℂ corresponds to the Lebesgue measure on V which is equal to 1 on the unit cube). Also for any subspace E let us denote by vol_{E} the Lebesgue measure on E which is equal to 1 on the unit cube. Under these identifications we get

𝔻 : (Val _{k}^{ev}(V))^{sm}˜⟶(Val _{n−k}^{ev}(V))^{sm}.

For this operator we have the following result which can be easily deduced from the last theorem.

Theorem 1.2.2. Let V be an n-dimensional Euclidean space. Then for any k = 0, 1, …, n

𝔻 : (Val _{k}^{ev}(V))^{sm}˜⟶(Val _{n−k}^{ev}(V))^{sm}

is an isomorphism and 𝔻^{2} = Id. This operator 𝔻 is defined uniquely by the following property: let ϕ ∈ Val _{k}^{ev}(V) and let E ∈ Gr_{k}(V); if ϕ|_{E} = f(E) · vol_{E} then 𝔻ϕ|_{E^{⊥}} = f(E)vol_{E^{⊥}}. Also 𝔻 commutes with the action of O(n).

Example. Let χ denote the Euler characteristic on a Euclidean space V. Clearly χ ∈ Val_{0}(V). Then 𝔻(χ) = vol_{V}, and 𝔻(vol_{V}) = χ.

2. Unitarily invariant valuations

In this section we will describe unitarily invariant translation invariant continuous valuations on convex compact subsets of ℂ^{n} by writing down explicitly a basis in this space. Let k, l be integers such that 0 ≤ k ≤ 2n and k/2 ≤ l ≤ n. Let us define a valuation

C_{k,l}(K) := ∫_{_{C}Gr_{l,n}} V_{k}(Pr_{F} (K))dF,

where the integration is with respect to the Haar measure on the complex Grassmannian of complex l-dimensional subspaces in ℂ^{n}, Pr_{F} denotes the orthogonal projection onto F, and V_{k}(Pr_{F} (K)) denotes the k-th intrinsic volume of Pr_{F} (K) inside F, namely it is the mixed volume of Pr_{F} (K) taken k times with the unit Euclidean ball in F taken 2l − k times. Clearly C_{k,l} ∈ Val _{k}^{U(n)}(ℂ^{n}). Note that for l = n we get the usual intrinsic volumes. For k = 0 we get the Euler characteristic, and for k = 2n, l = n we get the Lebesgue measure. Our next main result is:

Theorem 2.1.1. Let k be an integer, 0 ≤ k ≤ 2n. The dimension of the space Val _{k}^{U(n)}(ℂ^{n}) is equal to 1 + min{[k/2], [(2n − k)/2]}. The valuations C_{k,l} with \frac{max\{k,2n−k\}}{2} ≤ l ≤ n, form a basis of Val _{k}^{U(n)}(ℂ^{n}).

Remark. Later on in this section we will present another basis in the space of unitarily invariant valuations. This basis will be more convenient for the applications in integral geometry and for non-convex sets. In fact the connection between these two bases is not quite trivial and leads to new integral geometric formulas. This material will be discussed in more detail in Section 3.

Proof. The dimension of Val _{k}^{U(n)}(ℂ^{n}) was computed in [2]. Hence it remains to show that the valuations C_{k,l} with \frac{max\{k,2n−k\}}{2} ≤ l ≤ n are linearly independent. First of all it is clear that C_{k,l} = c · Λ(C_{k+1,l}), where Λ is the operator from the hard Lefschetz theorem (Theorem 1.1.1), and c is a nonzero constant depending on n, k, l only. Hence by the hard Lefschetz theorem for unitarily invariant valuations (Corollary 1.1.3) the statement is reduced to the case k ≥ n. Let us prove this case. We will prove the statement by induction in 2n − k. If 2n − k = 0 then the result is clear since by [24] any translation invariant continuous N-homogeneous valuation on ℝ^{N} is a Lebesgue measure. Now assume that n ≤ k < 2n, and the theorem is true for valuations homogeneous of degree > k. If k is odd then the induction assumption, Corollary 1.1.3, and the computation of the dimension of unitarily invariant k-homogeneous valuations imply the result. Hence let us assume that k is even. Again using Corollary 1.1.3 it is sufficient to check that C_{k,k/2} can not be presented as a linear combination of valuations C_{k,l} with l > \frac{k}{2}.

In order to prove it, we will show that the special orthogonal group SO(2n) acts differently on C_{k,k/2} and on C_{k,l} with l > \frac{k}{2}. To formulate this more precisely let us introduce some notation. First recall that the set of highest weights of SO(2n) is parameterized by sequences of integers μ_{1}, …, μ_{n−1}, μ_{n} such that μ_{1} ≥ ⋯ ≥ μ_{n−1} ≥ |μ_{n}|. For 1 ≤ l ≤ n let us denote by Λ(l) the subset of highest weights of SO(2n) such that all μ_{i}’s are even and if l < n satisfy in addition the following condition: μ_{j} = 0 for j > l.

The following result was proved in [2], Proposition 6.3.

Lemma 2.1.2. The natural representation of SO(2n) in the space Val _{k}^{ev}(ℂ^{n}) is multiplicity free and is isomorphic to a direct sum of irreducible components with highest weights (μ_{1}, μ_{2}, …, μ_{n}) ∈ Λ(min(k, 2n − k)) such that |μ_{2}| ≤ 2.

Note that the explicit description of the K-type structure was heavily based on the results of Howe and Lee [26].

The next result is well-known (see e.g., [46], §8).

Lemma 2.1.3. In every irreducible representation of SO(2n) the subspace of U(n)-invariant vectors is at most 1-dimensional. This subspace is 1-dimensional if and only if the highest weight of the irreducible representation of SO(2n) is of the form (μ_{1}, …, μ_{n}) where:

(i) if n is even then

μ_{1} = μ_{2} ≥ μ_{3} = μ_{4} ≥ ⋯ ≥ μ_{n−1} = μ_{n} ≥ 0;

(ii) if n is odd then

μ_{1} = μ_{2} ≥ μ_{3} = μ_{4} ≥ ⋯ ≥ μ_{n−2} = μ_{n−1} ≥ μ_{n} = 0.

The following lemma and Corollary 1.1.3 obviously imply Theorem 2.1.1.

Lemma 2.1.4. Let k be even, n ≤ k < 2n.

(i) The valuations C_{k,l} with l > k/2 belong to the sum of the representations with highest weights μ ∈ Λ(2n − k − 1).

(ii) The valuation C_{k,k/2} does not belong to the above sum.

Proof. First let us prove Part (i) of the lemma. As we have mentioned earlier C_{k,l} = c · Λ(C_{k+1,l}) if l > k/2. Since the operator Λ commutes with the action of SO(2n) on valuations then it is sufficient to check that C_{k+1,l} belongs to the sum of irreducible components with highest weights from Λ(2n − k − 1). As it was mentioned in Section 1 of this paper the space Val _{k+1}^{ev}(ℂ^{n}) can be imbedded into the space of continuous functions C(^{R}Gr_{k+1,2n}). But it is well-known that all irreducible representations of SO(2n) which appear in the last space belong to Λ(2n − k − 1) (see e.g., [46] §8 for the general case of compact symmetric spaces). This proves Part (i) of the lemma.

Let us prove Part (ii) which is somewhat more computational. We will show that the image of C_{k,k/2} in C(^{R}Gr_{k,2n}) is not orthogonal to the irreducible subspace in C(^{R}Gr_{k,2n}) with highest weight ( \underbrace{2, 2, …, 2}_{2n−k times} ,

0, …, 0). Clearly this will finish the proof of Lemma 2.1.4, and hence the proof of Theorem 2.1.1.

From the definition of C_{k,k/2} we immediately see that its image in C(^{R}Gr_{k,2n}) is the function f such that

f(E) = c · ∫_{_{C}Gr_{k/2,n}} | cos(E, F)|dF,

where c is a nonzero normalizing constant. In other words f is proportional to the cosine transform of the δ-function of the submanifold CGr_{k/2,n} ⊂ ^{R}Gr_{k,2n}. We will denote it by δ_{C}_{Gr}.

Lemma 2.1.5. Let k > n be even. Then δ_{C}_{Gr} is not orthogonal to the irreducible subspace in C(^{R}Gr_{k,2n}) with highest weight

( \underbrace{2, 2, …, 2}_{2n−k times} , 0, …, 0).

First let us deduce our statement from this lemma. The cosine transform commutes with the action of SO(2n) on C(^{R}Gr_{k,2n}). Hence by the Schur lemma it acts on each irreducible subspace as a multiplication by a scalar. Hence an irreducible subspace is contained in the image of the cosine transform if and only if the cosine transform on it does not vanish. However by Lemma 2.1.2 the irreducible subspace with the highest weight vector ( \underbrace{2, 2, …, 2}_{2n−k times} , 0, …, 0) is contained in the image

of Val _{2n,k}^{ev} in C(^{R}Gr_{k,2n}), and this image coincides with the image of the cosine transform by Theorem 1.1.3 of [6]. Thus it remains to prove Lemma 2.1.5 to finish the proof of Theorem 2.1.1.

Proof of Lemma 2.1.5. First observe that the statement of the lemma is purely representation theoretical. So replacing each subspace by its orthogonal complement we may and will assume that k ≤ n (oppositely to our previous assumption on k). Under this assumption it is easier to write down explicit formulas. It is sufficient to prove that δ_{C}_{Gr} is not orthogonal to the highest weight vector in the relevant irreducible subspace. This statement will be proven by a computation involving explicit form of the highest weight vector. First we will write it down following [45] (see also [22]).

Let e_{i,j} denote (2n × 2n)-matrix which has zeros at all but one place (i, j) where it has 1. Let us fix a Cartan subalgebra of so(2n) spanned by {C_{i}}_{i=1}^{n} where

C_{i} = e_{2i−1,2i} − e_{2i,2i−1}, i = 1, …, n.

For any subspace E ∈ ^{R}Gr_{k,2n} let us choose an orthonormal basis X^{1}, …, X^{k}, and let us write its coordinates in the standard basis in columns of 2n × k-matrix:

| \begin{matrix}X_{1}^{1} & & & … & & & X_{1}^{k} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ X_{2n}^{1} & & & … & & & X_{2n}^{k} \end{matrix} | .

Let X_{j} denote the j-th row of this matrix. For l ≤ n let A(l) be l × kmatrix whose j-th row is X_{2j−1}+\sqrt{−1}X_{2j}, j = 1, …, l. The next lemma was proved in [45], Theorem 5 (see also [22]).

Lemma 2.1.6. Let k ≤ n. Let (2m_{1}, 2m_{2}, …, 2m_{k}, 0, …, 0) be the highest weight of SO(2n) with m_{1} ≥ m_{2} ≥ ⋯ ≥ m_{k} ≥ 0. The irreducible subspace of C(^{R}Gr_{k,2n}) with this highest weight has the highest weight vector of the form

f_{m_{1},...,m_{k}} = det[A(1) · A(1)^{t}]^{m_{1}−m_{2}} · det[A(2) · A(2)^{t}]^{m_{2}−m_{3}} · ⋯ det[A(k) · A(k)^{t}]^{m_{k}}.

Recall that we are interested in the highest weight (\underbrace{2, 2, …, 2}_{k times},

0, …, 0). Hence the highest weight vector F ∈ C(^{R}Gr_{k,2n}) has the form:

F := det[A(k) · A(k)^{t}].

Let us denote for brevity m := k/2 (recall that m is an integer). We have to show that

∫_{M∈^{C}Gr_{m,n}} F (M)dM ≠ 0.

In fact we will show that the function F is nonnegative on ^{C}Gr_{m,n} and is not identically zero.

Let us choose in our hermitian space ℂ^{n} an orthonormal hermitian basis e_{1}, …, e_{n}. Then in the realization ℝ^{2n} of this space we will choose the basis (∗) e_{1}, e_{2}, e_{3}, …, e_{k}; \sqrt{−1}e_{1}, −\sqrt{−1}e_{2}, \sqrt{−1}e_{3}, …, −\sqrt{−1}e_{k}, other vectors.

Fix any E ∈ ^{C}Gr_{m,n}. Let us choose in E an orthonormal hermitian basis ξ_{1}, …, ξ_{m}. Then

ξ_{t} = ∑_{j=1}^{n} z_{t}^{j}e_{j} = ∑_{j=1}^{n}(Rez_{t}^{j} · e_{j} + Imz_{t}^{j} · (\sqrt{−1}e_{j})),

with z_{t}^{j} ∈ ℂ. Then the vectors ξ_{1}, …, ξ_{m}, \sqrt{−1}ξ_{1}, …, \sqrt{−1}ξ_{m} form a real basis of E. Let us write the coordinates of these vectors with respect to the basis (∗) in columns of the following matrix:

[| \begin{matrix}Rez_{1}^{1} & … & Rez_{m}^{1} & −Imz_{1}^{1} & … & −Imz_{m}^{1} \\ Rez_{1}^{2} & … & Rez_{m}^{2} & −Imz_{1}^{2} & … & −Imz_{m}^{2} \\ . . . . . . . . . . . . . . . . . . . . . & . . . . . . . . . . . . . . . . . . . . . . . . . \\ Rez_{1}^{k−1} & … & Rez_{m}^{k−1} & −Imz_{1}^{k−1} & … & −Imz_{m}^{k−1} \\ Rez_{1}^{k} & … & Rez_{m}^{k} & −Imz_{1}^{k} & … & −Imz_{m}^{k} \\ Imz_{1}^{1} & … & Imz_{m}^{1} & Rez_{1}^{1} & … & Rez_{m}^{1} \\ −Imz_{1}^{2} & … & −Imz_{m}^{2} & −Rez_{1}^{2} & … & −Rez_{m}^{2} \\ . . . . . . . . . . . . . . . . . . . . . & . . . . . . . . . . . . . . . . . . . . . . . . . \\ Imz_{1}^{k−1} & … & Imz_{m}^{k−1} & Rez_{1}^{k−1} & … & Rez_{m}^{k−1} \\ −Imz_{1}^{k} & … & −Imz_{m}^{k} & −Rez_{1}^{k} & … & −Rez_{m}^{k} \\ . . . . . . . . . . . . . . . . . . . . . & . . . . . . . . . . . . . . . . . . . . . . . . . \end{matrix} ] .

Now let us write down the (k × k)-matrix A(k). Recall that the j-th row of it is obtained by adding to (2j − 1)-th row of the above matrix i = \sqrt{−1} times the (2j)-th row of it. Then we obtain that A(k) is equal to

[\begin{matrix}Rez_{1}^{1} + iRez_{1}^{2} & … & Rez_{m}^{1} + iRez_{m}^{2} & −Imz_{1}^{1} − iImz_{1}^{2} & … & −Imz_{m}^{1} − iImz_{m}^{2} \\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \\ Rez_{1}^{k−1} + iRez_{1}^{k} & … & Rez_{m}^{k−1} + iRez_{m}^{k} & −Imz_{1}^{k−1} − iImz_{1}^{k} & … & −Imz_{m}^{k−1} − iImz_{m}^{k} \\ Imz_{1}^{1} − iImz_{1}^{2} & … & Imz_{m}^{1} − iImz_{m}^{2} & Rez_{1}^{1} − iRez_{1}^{2} & … & Rez_{m}^{1} − iRez_{m}^{2} \\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \\ Imz_{1}^{k−1} − iImz_{1}^{k} & … & Imz_{m}^{k−1} − iImz_{m}^{k} & Rez_{1}^{k−1} − iRez_{1}^{k} & … & Rez_{m}^{k−1} − iRez_{m}^{k} \end{matrix} ].

Let us denote by A the (m × m)-sub-matrix of the above matrix which stays in the upper left part of it, and by B the m × m-sub-matrix which stays in the lower left part of it. Then it is easy to see that

A(k) = [ \begin{matrix}A & −B \\ B & A \end{matrix} ] .

Then the function F is equal

det[A(k) · A(k)^{t}] = det[A(k)]^{2}.

Let us show that det[A(k)] ∈ ℝ. Indeed

\overline{det[A(k)]} = det [ \begin{matrix}A & −B \\ B & A \end{matrix} ] =

det ([ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] [ \begin{matrix}A & −B \\ B & A \end{matrix} ] [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ]) = det[A(k)].

It remains to show that F ≢ 0. Take E_{0} := span_{ℂ}{e_{1}, e_{3}, e_{5}, …, e_{k−1}}. Then F (E_{0}) = 1. q.e.d.

Now we will present another basis in the space of unitarily invariant valuations. As it was mentioned above this basis is more convenient to obtain integral geometric formulas for non-convex sets (see Section 3). Let ^{R}AGr_{k,2n} denote the Grassmannian of affine real k-dimensional subspaces in ℂ^{n} ≃ ℝ^{2n}. Let ^{C}AGr_{k,n} denote the Grassmannian of affine complex k-dimensional subspaces in ℂ^{n}. Note that ^{R}AGr_{k,2n} and ^{C}AGr_{k,n} have natural Haar measures which are unique up to a constant. For every nonnegative integers p and k such that 2p ≤ k ≤ 2n let us introduce the following valuations:

U_{k,p}(K) = ∫_{E∈^{C}AGr_{n−p,n}} V_{k−2p}(K ∩ E) · dE.

Clearly U_{k,p} ∈ Val _{k}^{U(n)}(ℂ^{n}).

Proposition 2.1.7. For any nonnegative integers k, p satisfying 2p ≤ k ≤ n one has

U_{k,p} = c_{n,k,p} · 𝔻(C_{2n−k,n−p}),

where c_{n,k,p} is a nonzero normalizing constant depending on n, k and p only. Hence the valuations U_{k,p} with 0 ≤ p ≤ \frac{min\{k,2n−k\}}{2} form a basis of the space Val _{k}^{U(n)}(ℂ^{n}).

Proof. Clearly the second statement immediately follows from the first one and Theorem 2.1.1. First we can rewrite the definition of U_{k,p} as follows:

U_{k,p}(K) = ∫_{F ∈^{C}Gr_{p,n}} dF · ∫_{x∈F} dx · V_{k−2p}(K ∩ (x + F^{⊥})),

where F^{⊥} denotes the orthogonal complement of F. Let us compute the image of U_{k,p} in the space C(^{R}Gr_{k,n}) under the imbedding described in Section 1. Fix any L ∈ ^{R}Gr_{k,n}. Let D_{L} denote the unit Euclidean ball inside L. Then by a straightforward elementary computation one can easily see that for K = D_{L} the inner integral in the last formula is equal to c · | cos(L, F)|, where c is a normalizing constant. Hence

(1)  U_{k,p}(D_{L}) = c · ∫_{F ∈^{C}Gr_{p,n}} dF · | cos(L, F)|
= c · ∫_{F ∈^{C}Gr_{p,n}} dF · | cos(L^{⊥}, F^{⊥})|
= c · ∫_{E∈^{C}Gr_{n−p,n}} dE · | cos(L^{⊥}, E)|.

It is easy to see that for any M ∈ ^{R}Gr_{k,2n}, and for 2k ≤ l

(2)  C_{k,l}(D_{M}) = c′ · ∫_{E∈^{C}Gr_{l,n}} dE · | cos(M, E)|,

where c′ is a normalizing constant. Clearly (1) and (2) imply the theorem. q.e.d.

3. Integral geometry in ℂ^{n}

Using the classification of unitarily invariant valuations obtained in the previous section, we will establish new integral geometric formulas for real submanifolds in ℂ^{n}. Note that these formulas will be valid not only for convex domains, but for arbitrary piecewise smooth submanifolds of ℂ^{n} with corners.

The method to obtain the result for non-convex sets using the convex case is as follows. First one should guess the correct formula for the general case. Next one can approximate nicely piecewise smooth set by polyhedral sets. The last set can be presented as a finite union of convex polytopes. For each convex polytope and for each finite intersection of them we can apply the formulas for the convex case. The final result follows by the inclusion-exclusion principle. In Subsection 3.1 we obtain new integral geometric formulas in ℂ^{n}. In Subsection 3.2 we compute explicitly the constants in one of these formulas in the particular case n = 2. In Subsection 3.3 we discuss another example of unitarily invariant valuation, Kazarnovskii’s pseudovolume.

3.1 General results

Let us denote by IU(n) the group of all isometries of ℂ^{n} preserving the complex structure. (Clearly this group is isomorphic to the semidirect product ℂ^{n} ⋊ U(n).)

Note also that the intrinsic volumes V_{i} in a Euclidean space ℝ^{N} can be defined not only for convex compact domains but also for compact domains with piecewise smooth boundary (even more generally, for compact piecewise smooth submanifolds with corners). For instance for a domain Ω with smooth boundary they can be defined as follows: V_{i}(Ω) := \frac{1}{N} M_{N−1−i}(∂Ω), where for any hypersurface Σ

M_{r}(Σ) := \binom{N − 1}{r}^{−1} ∫_{Σ}\{k_{i_{1}}, …, k_{i_{r}}\}dσ,

where {k_{i_{1}}, …, k_{i_{r}}} denotes the r-th elementary symmetric polynomial in the principal curvatures k_{i_{1}}, …, k_{i_{r}}, and dσ is the measure induced by the Riemannian metric.

Then we can define the expressions U_{k,p}(Ω) for 0 ≤ 2p ≤ k ≤ 2n (for convex compact sets they were defined in Section 2). The correct generalization is as follows:

U_{k,p}(Ω) = ∫_{E∈^{C}AGr_{n−p,n}} V_{k−2p}(Ω ∩ E) · dE,

where we use the above definition of V_{k−2p}(Ω).

Remark. In fact the expressions U_{k,p} can be defined also for compact piecewise smooth submanifolds of ℂ^{n} with corners.

Theorem 3.1.1. Let Ω_{1}, Ω_{2} be compact domains in ℂ^{n} with piecewise smooth boundaries such that for every U ∈ IU(n) the intersection Ω_{1} ∩ U(Ω_{2}) has finitely many components. Then

∫_{U∈IU(n)} χ(Ω_{1} ∩ U(Ω_{2}))dU

= ∑_{k_{1}+k_{2}=2n} ∑_{p_{1},p_{2}} κ(k_{1}, k_{2}, p_{1}, p_{2})U_{k_{1},p_{1}}(Ω_{1})U_{k_{2},p_{2}}(Ω_{2}),

where the inner sum runs over 0 ≤ p_{i} ≤ \frac{min\{k_{i},2n−k_{i}\}}{2}, i = 1, 2, and κ(k_{1}, k_{2}, p_{1}, p_{2}) are certain constants depending on n, k_{1}, k_{2}, p_{1}, p_{2} only.

Theorem 3.1.2. Let Ω be a compact domain in ℂ^{n} with piecewise smooth boundary. Let 0 < q < n, 0 < 2p < k < 2q. Then

∫_{E∈^{C}AGr_{q,n}} U_{k,p}(Ω ∩ E) = ∑_{p=0}^{[k/2]+n−q} γ_{p} · U_{k+2(n−q),p}(Ω),

where the constants γ_{p} depend only on n, q, and p.

Let us denote by ALGr_{n} the (noncompact) Grassmannian of affine Lagrangian subspaces of ℂ^{n}. Clearly it is a homogeneous space of the group IU(n).

Theorem 3.1.3. Let Ω be a compact domain in ℂ^{n} with piecewise smooth boundary. Then

∫_{ALGr(ℂ^{n})} χ(E ∩ Ω)dE = ∑_{p=0}^{[n/2]} β_{p} · U_{n,p}(Ω),

where β_{p} are certain constants depending on n and p only.

Remarks.

1) Theorems 3.1.1 and 3.1.2 are analogs of general kinematic formulas of Poincaré, Chern [11], [13] and Federer [15] (see also [43], especially Ch. 15).

2) These results can be formulated and proved not only for domains but also for piecewise smooth compact submanifolds in ℂ^{n} with corners. To do it, consider an ε-neighborhood of this submanifold for small ε > 0. Then apply the above formulas to this domain. Both sides depend polynomially on ε. Comparing the lowest degree terms we get the mentioned generalizations. We do not reproduce here the explicit computations.

3) It would be of interest to compute the constants κ(k_{1}, k_{2}, p_{1}, p_{2}) and β_{p} in Theorems 3.1.1, 3.1.2, and 3.1.3 explicitly. We could not do it in general. But we compute them in the first nontrivial case n = 2 for Theorem 3.1.3 in the next subsection.

3.2 Integral geometry in ℂ^{2}

In this subsection we will compute explicitly the constants in one of the integral geometric formulas discussed in the previous subsection in the particular case of ℂ^{2}. In order to do these computations we first recall the classical presentations for the orthogonal group SO(4) (more precisely for its universal covering Spin(4)) and for the Grassmannian of oriented 2-planes in ℝ^{4} which we will denote by ^{R}Gr_{2,4}^{+}.

Let us denote the standard complex structure on ℂ^{2} by i. Let us identify ℂ^{2} with the quaternionic space ℍ with the usual anti-commut-ing complex structures i, j, and k = ij. Then clearly ℍ = ℂ ⊕ jℂ.

Recall that the group of quaternions with norm equal to 1 acts by left multiplication on ℍ ≡ ℂ^{2} and thus is identified with the group SU(2).

Also we have the isomorphism

Φ : SU(2) × SU(2)/{±Id}˜⟶SO(4)

defined by

Φ((q_{1}, q_{2}))(x) = q_{1}xq_{2}^{−1},

where q_{1}, q_{2} are norm one quaternions. Hence we can and will identify the group Spin(4) with SU(2) × SU(2). Let E_{0} ∈ ^{R}Gr_{2,4}^{+} be the span_{ℝ}{1, i} with standard orientation coming from the complex structure. Clearly the stabilizer of E_{0} in Spin(4) = SU(2) × SU(2) is equal to T × T where

T = {z ∈ ℂ||z| = 1} = U(1) ⊂ SU(2).

Hence we have the following presentation of the Grassmannian of real oriented 2-planes in ℝ^{4} :

RGr_{2,4}^{+} = SU(2)/T × SU(2)/T.

However SU(2)/T ≃ ℂP^{1}, where ℂP^{1} is (as usual) the complex projective line. For our computations it will be convenient to identify ℂP^{1} with the 2-dimensional sphere of radius 1/2. Moreover it will be convenient to consider this sphere S^{2} in the standard coordinate space ℝ^{3} with the center (1/2, 0, 0). Moreover E_{0} ∈ ^{R}Gr_{2,4}^{+} = S^{2} × S^{2} will correspond to the point ((1, 0, 0), (1, 0, 0)). The following lemma can be proved by a straightforward computation.

Lemma 3.2.1. Let E = (t_{1}, t_{2}) ∈ S^{2} × S^{2} = ^{R}Gr_{2,4}^{+}. Let t_{i} = (x_{i}, y_{i}, z_{i}), i = 1, 2. Then | cos(E, E_{0})| = |x_{1} + x_{2} − 1|.

For ℂ^{2} Theorem 2.1.1 says:

Proposition 3.2.2. For 0 ≤ k ≤ 4, k ≠ 2, the space Val _{k}^{U(2)}(ℂ^{2}) is spanned by V_{k}; Val _{2}^{U(2)}(ℂ^{2}) is spanned by V_{2} and by ϕ, where ϕ(K) = ∫_{ξ∈CP^{1}} vol_{2}(Pr_{ξ}K)dξ.

Recall that the total measure of ℂP^{1} is chosen to be equal to one. Now let us describe the image of the valuation ϕ in C(^{R}Gr_{2,4}^{+}). Let us denote this image by f.

Lemma 3.2.3. For every E = (t_{1}, t_{2}) ∈ S^{2} × S^{2} = ^{R}Gr_{2,4}^{+} with t_{i} = (x_{i}, y_{i}, z_{i}), i = 1, 2

f(E) = vol_{2}D_{2} · ((x_{2} − \frac{1}{2})^{2} + \frac{1}{4}) ,

where D_{2} denotes the unit 2-dimensional Euclidean disk.

This lemma follows immediately from Lemma 3.2.1 and the fact that the set of complex lines in ℂ^{2} is SU(2)-orbit of E_{0}.

Let us denote by LGr_{n} the Grassmannian of Lagrangian subspaces in ℂ^{n}. Let us define a valuation ψ ∈ Val _{2}^{U(2)}(ℂ^{2}) as follows:

ψ(K) = ∫_{F ∈LGr_{2}} vol_{2}(Pr_{F} (K))dF,

where dF is the Haar measure on LGr_{2} normalized by 1. The main result of this subsection is:

Theorem 3.2.4.

ϕ + 2ψ = \frac{π}{V_{2}(D_{2})}V_{2}.

Proof. Let

ϕ̃:= \frac{1}{vol_{2}D_{2}} ϕ = \frac{1}{π} ϕ, ψ̃:= \frac{1}{vol_{2}D_{2}} ψ = \frac{1}{π} ψ.

Let us denote by g̃ the image of ψ̃ in C(^{R}Gr_{2,4}^{+}), and by f̃ the image of ϕ̃ in C(^{R}Gr_{2,4}^{+}). By Lemma 3.2.3

(∗∗)  f̃(E) = (x_{2} − \frac{1}{2})^{2} + \frac{1}{4}

for every E = (t_{1}, t_{2}) ∈ S^{2} × S^{2} = ^{R}Gr_{2,4}^{+} with t_{i} = (x_{i}, y_{i}, z_{i}), i = 1, 2. Now let us describe g̃. Thus f̃ is considered as a function on the second copy of S^{2}. Let E_{1} = span{1, j} ∈ LGr_{2}. It is easy to see that E_{1} = U_{0}(E_{0}), where U_{0} ∈ SO(4) is defined by

U_{0}(x) = \frac{1 + k}{\sqrt{2}} · x · \overline{\frac{1 + k}{\sqrt{2}}}

for every x ∈ ℍ = ℝ^{4}. Then

g̃(E) = ∫_{F ∈LGr_{2}} | cos(F, E)|dF
= ∫_{U∈U(2)} | cos(U(E_{1}), E)|dU
= ∫_{U∈U(2)} | cos(UU_{0}(E_{0}), E)|dU
= ∫_{U∈U(2)} | cos(E_{0}, U_{0}^{−1}U(E))|dU.

However U(2) = (SU(2)×U(1))/{±1}, where (q, λ) ∈ SU(2)×U(1) acts on x ∈ ℍ by x ↦ q · x · λ^{−1}. In the formulas below we will write the action of λ ∈ U(1) on E ∈ ^{R}Gr_{2,4}^{+} from the right: λ(E) = E · λ^{−1}. In this notation the last integral can be rewritten as

∫_{V ∈SU(2)} dV ∫_{λ∈U(1)} dλ · |cos (E_{0}, \frac{1 − k}{\sqrt{2}} V · E · λ^{−1} · \frac{1 + k}{\sqrt{2}} )|

= ∫_{V ∈SU(2)} dV ∫_{λ∈U(1)} dλ · |cos (E_{0}, V · E · λ^{−1} \frac{1 + k}{\sqrt{2}} )|

= ∫_{λ∈U(1)} dλ · f̃(E · λ^{−1} · \frac{1 + k}{\sqrt{2}} ) .

By (∗∗) we can write f̃ as

f̃(E) = h(E) + \frac{1}{3},

where h(E) = (x_{2} − \frac{1}{2})^{2} − \frac{1}{12}. The function h on the sphere S^{2} (of radius 1/2) has the property

∫_{S^{2}} h = 0.

We have

g̃(E) = ∫_{λ∈U(1)} dλ · h (E · λ^{−1} · \frac{1 + k}{\sqrt{2}} ) + \frac{1}{3}.

It is easy to see that

h_{1}(E) := h (E · \frac{1 + k}{\sqrt{2}} ) = y_{2}^{2} − \frac{1}{12},

−1/2 ≤ y_{2} ≤ 1/2. Clearly h_{1} is a polynomial of second degree on sphere S^{2} such that ∫_{S^{2}} h_{1} = 0. Hence ∫_{λ∈U(1)} h_{1}(E ·λ^{−1}) dλ also satisfies these properties, and moreover it is U(1)-invariant. But such a polynomial is unique up to proportionality, hence ∫_{λ∈U(1)} h_{1}(E · λ^{−1}) dλ = c · h(E), where c is a constant. Let us compute it. If subspace E is such that x_{2} = 1/2 then h(E) = −1/12. But

∫_{λ∈U(1)} h_{1}(E · λ^{−1}) dλ = \frac{1}{2π} ∫_{0}^{2π} dϕ (\frac{cos^{2} ϕ}{4} − \frac{1}{12}) = 1/24.

Hence c = −1/2. Thus g̃(E) = −\frac{h(E)}{2} + \frac{1}{3} = −\frac{1}{2}(f̃(E)− \frac{1}{3})+ \frac{1}{3} = \frac{1}{2} − \frac{f̃(E)}{2}. Hence ϕ̃ + 2ψ̃ = κ · V_{2}, where κ is a normalizing constant such that for the unit 2-dimensional Euclidean disk D_{2}, κ · V_{2}(D_{2}) = 1. Thus we get that ϕ + 2ψ = \frac{π}{V_{2}(D_{2})}V_{2}. q.e.d.

3.3 Kazarnovskii’s pseudovolume

In this subsection we discuss another example of unitarily invariant translation invariant continuous valuation which has rather different origin, namely it comes from complex analysis. We discuss so called Kazarnovskii’s pseudovolume. The main result of this subsection is a new formula for Kazarnovskii’s pseudovolume in integral geometric terms. The proof of this result is based on the classification of unitarily invariant valuations (Theorem 2.1.1).

Now let us recall the definition of Kazarnovskii’s pseudovolume following [30], [31]. Let ℂ^{n} be Hermitian space with the Hermitian scalar product (·, ·). For a convex compact set K ∈ K(ℂ^{n}) let us denote its supporting functional

h_{K}(x) := sup_{y∈K}(x, y).

For a set K ∈ K(ℂ^{n}) such that its supporting functional h_{K} is smooth on ℂ^{n} − {0} Kazarnovskii’s pseudovolume P is defined as follows:

P (K) := ∫_{D}(dd^{c}h_{K})^{n},

where D denotes the unit Euclidean ball on ℂ^{n}, and d^{c} = I^{−1} ∘ d ∘ I for our complex structure I.

Proposition 3.3.1. Kazarnovskii’s pseudovolume P extends by continuity in the Hausdorff metric to all K(ℂ^{n}). Then P is unitarily invariant translation invariant continuous valuation homogeneous of degree n.

Proof. The first part of the proposition (the continuity) is a standard fact from the theory of plurisubharmonic functions originally due to Chern-Levine-Nirenberg [14] (see also [30], [31]). The unitary invariance, translation invariance, and the homogeneity of degree n are obvious. The only thing which remains to prove is that P is a valuation.

Let A be a convex polytope. It was shown by Kazarnovskii [30] that

P (A) = κ ∑_{F} f(F)γ(F)vol_{n}F,

where κ is a normalizing constant, the sum runs over all n-dimensional faces F of A, γ(F) is the measure of the exterior angle of A at F, vol_{n}F denotes the (n-dimensional) volume of the face F, and f(F) is defined as follows. Let D_{F} denote the unit ball in the linear subspace parallel to the face F. Then f(F) = vol(D_{F} + I · D_{F}). It is easy to see from the above formula that P restricted to the class of convex compact polytopes is a valuation, namely if A_{1}, A_{2}, A_{1} ∪ A_{2} are convex compact polytopes then

P (A_{1} ∪ A_{2}) = P (A_{1}) + P (A_{2}) − P (A_{1} ∩ A_{2}).

Then it is easy to see that the continuity of P and the valuation property on the subclass of polytopes imply that P is a weak valuation on K(ℂ^{n}) (this means that for any real hyperplane H and any K ∈ K(ℂ^{n}) one has P (K) = P (K ∩ H^{+}) + P (K ∩ H^{−}) − P (K ∩ H) where H^{+} and H^{−} denote the half-spaces). However it was shown by Groemer [23] that every continuous weak valuation is valuation (in the usual sense). q.e.d.

The main result of this subsection is as follows.

Theorem 3.3.2.

P = ∑_{n/2≤l≤n} α_{l}C_{n,l},

where α_{l} ∈ ℝ are certain constants depending only on n, and C_{n,l} are valuations defined in the previous section.

Remark. It would be interesting to compute explicitly the constant α_{l}.

Proof. The proof follows immediately from Proposition 3.3.1 and Theorem 2.1.1. q.e.d.

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Department of Mathematics Tel Aviv University, Ramat Aviv

69978 Tel Aviv, Israel